Question

In a recent​ year, a poll asked 2360 random adult citizens of a large country how they rated economic conditions. In the​ poll, 28​% rated the economy as​ Excellent/Good. A recent media outlet claimed that the percentage of citizens who felt the economy was in​ Excellent/Good shape was 25​%. Does the poll support this​ claim? ​a) Test the appropriate hypothesis. Find a 99​% confidence interval for the proportion of adults who rated the economy as​ Excellent/Good. Check conditions. ​b) Does your confidence interval provide evidence to support the​ claim? ​c) What is the significance level of the test in​ b? Explain. ​a) Let p be the proportion of adult citizens who felt the economy was in​ Excellent/Good shape. What are the null and alternative hypotheses for the​ test? A. H0​: pequals0.28 vs. HA​: pless than0.28 B. H0​: pequals0.25 vs. HA​: pnot equals0.25 C. H0​: pequals0.28 vs. HA​: pnot equals0.28 D. H0​: pequals0.25 vs. HA​: pgreater than0.25 Check the conditions. Which of the following conditions are​ satisfied? Select all that apply. A. Less than​ 10% of the population was sampled. B. The data are independent. C. There are more than 10​ "successes" and 10​ "failures." D. The sample is random. Find a 99​% confidence interval for the proportion of adults who rated the economy as​ Excellent/Good. Select the correct choice below and fill in any answer boxes within your choice. A. left parenthesis nothing comma nothing right parenthesis ​(Round to three decimal places as​ needed.) B. The conditions of a confidence interval are not satisfied. ​b) Does your confidence interval provide evidence to support the​ claim? A. Yes. Since pequals0.25 is within the​ interval, there is evidence to support the claim. B. Yes. Since pequals0.25 is not within the​ interval, there is evidence to support the claim. C. No. Since pequals0.25 is within the​ interval, there is evidence that the proportion is not p equals 0.25. D. No. Since pequals0.25 is not within the​ interval, there is evidence that the proportion is not p equals 0.25. E. The confidence interval cannot be found because not all of the conditions are satisfied. ​c) The significance level is alphaequals 34​, because the test is a two ​-sided test and is based on a 99​% confidence interval.

Answer 1

Here we have given that,

n=Total number of adults citizens of a large country =2360

Now, we estimate the sample proportion as

=sample proportion of adults citizens of a large country rated the economy as excellent/good

=28% i.e. 0.28

p=populaiton proportion of the citizens who felt the economy was in excellent/good shape=25% i.e. 0.25

(a)

The below-mentioned conditions are satisfied for constructing the CI,

1. Each observation is independent to each other the sample is random.
2. n*p=2360*0.25=590 and n*p*(1-p) = 2360*0.24*(1-0.25)=443 both are greater than 10. i.e we can say that there are more than 10 success and failures.
3. less than 10% of the population was sampled.

The level of significance is as follows,

c=confidence level =0.99

=level of significance= 1-c=1-0.99=0.01

Claim: To check whether the populaiton percentage of citizens who felt the economy was in Excellent/Good shape was 25%.

The null and alternative hypothesis is as follows,

Versus

i.e. option B is correct.

Now we want to find the 99% confidence interval for the population proportion of the citizens who felt the economy was in excellent/good shape p

The formula for CI is as follows,

Now, first, we can find two-tailed Z-critical value

This 99% CI is two-tailed.

c=confidence level =0.99

=level of significance= 1-c=1-0.99=0.01

                            = Zcritical (0.005)

=2.58 Using EXCEL software =ABS(NORMSINV(probability =0.005))

The 99% confidence interval is,

The 99% confidence interval for p is (0.256, 0.304)

Interpretation:

Here we can say that we are 99% confident that the populaiton proportion of the citizens who felt the economy was in excellent/good shape will fall inside this interval.

Conclusion regarding the claim (Hypothesis):

No, 95% confidence interval support the claim, Since p = 0.25 is not within the interval, there is evidence that the proportion is not equal to 0.25.

i.e. here option D is correct.